The one-dimensional (1D) theoretical models for thermoelectric generators (TEGs) widely used to interpret experimental results and guide device design have not been subjected to a rigorous re-evaluation. In this work, we examine the foundations of the traditional 1D TEG model and identify the omission of the heat flow induced by the Seebeck effect, and inconsistencies in boundary conditions, both of which can lead to deviations in predicting the origin of thermoelectric power and the energy conversion efficiency. A new 1D model with Seebeck heat flow and four junction temperatures is proposed based on actual thermoelectric generators. In addition, The TEG system is decomposed into a thermal subsystem and an electrical subsystem, which can clarify the energy conservation relation at the interface and inside element. Numerical solutions to the governing equations system of the new 1D model show that compared to traditional 1D models, the new 1D model has three advantages: (1) the temperature at both ends of the thermoelectric element is variable, not fixed; (2) the thermoelectric conversion efficiency can be more accurately predicted; (3) the output electrical power does not come from the conversion of Peltier heat flow, but from the conversion of heat flow provided by the heat reservoir.

At present, the advancement of thermoelectric technology remains largely focused on developing high-performance thermoelectric materials, while comparatively little attention is directed towards its fundamental principles. To address this gap, this study introduces a new physical quantity, the “thermoelectric diffusion potential”, which clarifies the physical interpretations of various thermoelectric coefficients. Analyses reveal that, within a thermoelectric element, the Seebeck coefficient represents a balance between the thermoelectric diffusion field and electrostatic field, rather than between temperature and voltage differences. Using the thermoelectric diffusion potential, the relationship between the Seebeck and Peltier coefficients can be derived directly. Building on this framework, two additional physical quantities, namely the “thermoelectric energy” and “thermoelectric energy flow”, associated with the thermoelectric diffusion potential, are introduced. The formulation of thermoelectric energy flow helps derive the energy conversion relationship at the interface on a macroscopic level. Specifically, energy conversion at the interface occurs between thermoelectric and thermal energy flows, while within the element, it takes place between thermoelectric and electrical energy flows. Owing to the dual nature of internal energy in thermoelectric materials, manifesting as both thermal and electrical energy, the conversion within the element can also be regarded as one between thermal and electrical energy flows. The proposed quantities constitute an important complementary interpretation for the existing thermoelectric framework.

Since the ideal gas equation of state (EOS) was established in 1840, a wide variety of EOS theories have been developed. However, due to the diversity of material structures and the complexity of intermolecular interactions, numerous EOS either have complex forms or have empirical coefficients without physical meaning, which severely limits their applications. This paper builds a simple and universal EOS model by means of a fully macroscopic thermodynamic approach. Firstly, two single variable thermodynamic functions as a function of pressure only and as a function of temperature only, respectively, are constructed. On this basis, two EOS in the forms of P–V–T and P–S–T are obtained by thermodynamic derivation, which are almost as simple as the ideal gas EOS. There are no assumptions about material structures and intermolecular interactions involved here. Therefore, the model is universal. Moreover, the coefficients in these two EOS have clear thermodynamic significance and thus can be calculated directly without fitting. The model is shown to characterize the thermodynamic properties of substances well and may play an important role in high-density and supercritical applications. This work may provide a new way of developing EOS theory and enrich the fundamentals of thermodynamics.

The research on thermoelectric materials has flourished over the past half-century; however, the current theoretical understanding of thermoelectric power generation remains underdeveloped. After clarifying that thermoelectric power generation is not a cyclic process, we propose an alternative framework based on a combination of the heat and electricity conductive processes involving heat sources and sinks. We formulate and solve a system of 21 governing equations, which successfully describes the thermoelectric power generation process. The calculated results for load output power are consistent with both existing theories and experimental data, which indicates that the thermoelectric conversion in TEGs is not achieved through the cycle process of the working fluid, but through the conductive processes of heat and electricity. In addition, our results reveal that the thermoelectric conversion efficiency does not monotonically increase with the Seebeck coefficient, if the Seebeck coefficient is very large; rather, it reaches a maximum value at a specific point. The location of this maximum efficiency depends on the thermal and electrical resistances in the power generation circuit. This phenomenon only has theoretical significance at present.

With the increasingly widespread application of thermoelectric cooling technology, the demand for precise characterization of the Peltier effect at material heterointerfaces is growing rapidly. It can help us develop thermoelectric materials and study the charge–phonon interaction and energy conversion laws at heterointerfaces. However, direct measurement results for Peltier heat are scarce, mainly because existing electrical or thermal imaging measurement methods find it difficult to accurately measure the heat flow through samples. Furthermore, the Peltier effect in thermoelectric elements occurs simultaneously with several heat generation and heat transfer processes, adding further complexity to the measurement. In this study, we proposed a Peltier effect characterization method based on infrared thermography and accurately characterized the Peltier heat of the Bi2Te3 and Mg3(Sb,Bi)2 PN junction heterointerface using a steady-state and direct experimental system. The measurement results can form a good mutual verification with the Kelvin relationship and the Onsager reciprocal relationship. Compared with other existing measurement techniques, our method quantitatively analyzes the value of radiation heat, eliminates its influence by bidirectional current method, and can directly obtain the value of heat flux. This measurement method can achieve a temperature resolution of 50 mK and a heat flux resolution of 0.1 mW, which not only presents an effective means of accurately measuring small heat flux but also provides guidance for the development of thermoelectric theory and thermoelectric devices.

Bridgman once reflected on thermodynamics that the laws of thermodynamics were formulated in their present form by the great founders of thermodynamics, Kelvin and Clausius, before all the essential physical facts were in, and there has been no adequate reexamination of the fundamentals since. Thermodynamics still has unknown possibilities waiting to be explored. This paper begins with a brief review of Clausius’s work on the second law of thermodynamics and a reassessment of the content of Clausius’s statement. The review tells that what Clausius originally referred to as the second law of thermodynamics was, in fact, the theorem of equivalence of transformations (TET) in a reversible cycle. On this basis, a new symmetric form of Clausius’s TET is proposed. This theorem says that the two transformations, i.e., the transformation of heat to work and the transformation of work from high pressure to low pressure, should be equivalent in a reversible work-to-heat cycle. New thermodynamic cyclic laws are developed on the basis of the cycle with two work reservoirs (two pressures), which enriches the fundamental of the second law of thermodynamics.

The Kelvin relation, relating the Seebeck coefficient and the Peltier coefficient, is a theoretical basis of thermoelectricity. It was first derived by Kelvin using a quasi-thermodynamic approach. However, Kelvin’s approach was subjected to much criticism due to the rude neglect of irreversible factors. It was only later that a seemingly plausible proof of the Kelvin relation was given using the Onsager reciprocal relation with full consideration of irreversibility. Despite this, a critical issue remains. It is believed that the Seebeck and Peltier effects are thermodynamically reversible, and therefore, the Kelvin relation should also be independent of irreversibility. Kelvin’s quasi-thermodynamic approach, although seemingly irrational, may well have touched on the essence of thermoelectricity. To avoid Kelvin’s dilemma, this study conceives the physical scenarios of equilibrium thermodynamics to explore thermoelectricity. Unlike Kelvin’s quasi-thermodynamic approach, here, a completely reversible thermodynamic approach is used to establish the reciprocal relations of thermoelectricity, on the basis of which the Kelvin relation is once again derived. Moreover, a direct thermodynamic derivation of the Onsager reciprocal relations for fluxes defined as the time derivative of an extensive state variable is given using the method of equilibrium thermodynamics. The present theory can be extended to other coupled phenomena.

Here, we investigate the maximum power and efficiency of thermoelectric generators through devising a set of protocols for the isothermal and adiabatic processes of thermoelectricity to build a Carnot-like thermoelectric cycle, with the analysis based on fluctuation theorem. The Carnot efficiency can be readily obtained for the quasistatic thermoelectric cycle with vanishing power. The maximum power-efficiency pair of the finite-time thermoelectric cycle is derived, which is found to have the identical form to that of Brownian motors characterized by the stochastic thermodynamics. However, it is of significant discrepancy compared to the linear-irreversible and endoreversible-thermodynamics based formulations. The distinction with the linear-irreversible-thermodynamics case could result from the difference in the definitions of Peltier and Seebeck coefficients in the thermoelectric cycle. As for the endoreversible thermodynamics, we argue the applicability of endoreversibility could be questionable for analyzing the Carnot-like thermoelectric cycle, due to the incompatibility of the endoreversible hypothesis that attributes the irreversibility to finite heat transfer with thermal reservoirs, though the distinction in the mathematical expressions can vanish with the assumption that the ratio of thermoelectric power factors at the high and low temperatures (γ) is equal to the square root of the temperature ratio, γ=sqrt[T_{L}/T_{H}] (this condition could significantly deviate from the practical case). Last, utilizing our models as a concise tool to evaluate the maximum power-efficiency pairs of realistic thermoelectric material, we present a case study on the n-type silicon.

Current research on thermoelectricity is primarily focused on the exploration of materials with enhanced performance, resulting in a lack of fundamental understanding of the thermoelectric effect. Such circumstance is not conducive to the further improvement of the efficiency of thermoelectric conversion. Moreover, available physical images of the derivation of the Kelvin relations are ambiguous. Derivation processes are complex and need a deeper understanding of thermoelectric conversion phenomena. In this paper, a new physical quantity 'thermoelectrical potential' from the physical nature of the thermoelectric conversion is proposed. The quantity is expressed as the product of the Seebeck coefficient and the absolute temperature, i.e., ST. Based on the thermoelectrical potential, we clarify the conversion of the various forms of energy in the thermoelectric effect by presenting a clear physical picture. Results from the analysis of the physical mechanism of the Seebeck effect indicate that the thermoelectrical potential, rather than the temperature gradient field, exerts a force on the charge carriers in the thermoelectric material. Based on thermoelectric potential, the Peltier effects at different material interfaces can be macroscopically described. The Kelvin relation is rederived using the proposed quantity, which simplified the derivation process and elucidated the physical picture of the thermoelectrical conversion.

Heat conduction optimization with arbitrary boundary conditions is a challenging problem that lacks a universal optimization criterion. In the present work, the concept of generalized entransy dissipation (GED) is proposed through transforming heat conduction optimization problems with arbitrary boundaries into their homogeneous counterparts. It is demonstrated that minimizing GED leads to optimal thermal performance of heat conduction problems with arbitrary boundary conditions. In addition, GED-based continuous optimization problems are convex, guaranteeing the uniqueness and global optimality of the solution and benefitting numerical calculations. Two typical problems with complex boundary conditions are studied by applying the minimum principle of GED, and the results are compared with other optimization objectives. The numerical results show that GED achieves better thermal performance than entropy generation-(EG) and entransy dissipation-(ED) based optimizations. For the optimization of boundary average temperature under the given input heat flux of 200 W, GED achieves the best result, where the optimized average temperature is 48.8 K and 27.5 K lower compared with EG and ED optimizations, respectively. In general, GED offers a reasonable and easy to implement optimization principle for heat conduction processes with arbitrary boundaries and may provide new insights for heat conduction optimization.

According to the analogy between convection and conduction with heat sources, three ways are used to enhance convective heat transfer, including the increase of Re and Pr numbers, uniformity of velocity and temperature profiles, and the included angles of velocity and temperature vectors. With the latter two ways, further understanding can be obtained for the conventional heat transfer enhancement techniques on the one hand and some new approaches of heat transfer enhancement techniques have been suggested on the other hand. Analytical and experimental studies show that the enhancement approaches aimed at improving the coordination of flow and temperature fields or direct improvement of uniformity of fluid temperature profile can remarkably enhance heat transfer with less additional pressure drop.

Ideal gas is the most fundamental and simple system in thermodynamics, which has extensive applications in energy research and engineering. By reviewing the physical concept of ideal gas, it is found that the current understanding of ideal gas is still inappropriate and ambiguous, making it challenging to reveal the essential difference between ideal and real gases. Therefore, the macro and microscopic definitions and properties of ideal gas need to be reconceptualized. On the microscopic level, an ideal gas is a hypothetical collection of classical masses in irregular motion, which does not involve quantum mechanics; on the macroscopic level, any real gas cannot strictly obey the equation of state of ideal gas. Moreover, the heat capacity is a constant which should be an endogenous attribute of ideal gas model in order to unify the macroscopic and microscopic definitions of ideal gas. Finally, according to the law of heat capacity, the real gas can be divided into three categories: far-ideal, near-ideal, and quasi-ideal. This makes it easier to perform thermodynamic calculations with the assistance of ideal gas. Among them, the far-ideal gas heat capacity is a function of temperature and pressure, the near-ideal gas heat capacity is a single-valued function of temperature, and the quasi-ideal gas heat capacity is a constant.

The assumption of flow continuum can usually be accepted for flow in MEMS with channels from tens of micrometer (Kn~0.001) to fractions of 1 millimeter (Kn~0.0001). Hence, the mechanism for the departure of flow and heat transfer correlations for MEMS from standard ones can be largely attributed to the variation of dominant factors in flow and heat transfer when the scale goes down. Both analytical and experimental results show that the compressibility for the gas flow in the micro channels can be very remarkable. And it leads to the increase of friction factor and the Nusselt number. The surface roughness should be another factor which is responsible for the different flow and heat transfer behavior in MEMS. Experimental results show that the flow behavior in the micro channels for the smooth tubes is very close to usual one, but changes remarkably even when the relative roughness is 2-4 % only. As the object size going down, the dominant forces will vary, for example, the viscous force is dominant over the inertial force for natural convection at small Grashof number, and consequently, the balance between the buoyancy and viscous force leads to a heat transfer correlation, Nu~Gr1/3, which differs from standard one.

In 2019, Schilling et al. claimed that they achieved the supercooling of a body without external intervention in their thermoelectric experiments, thus arguing that the second law of thermodynamics was bent. Kostic suggested that their claim lacked full comprehension of the second law of thermodynamics. A review of history shows that what Clausius referred to as the second law of thermodynamics is the theorem of the equivalence of transformations (unfairly ignored historically) in a reversible heat–work cycle, rather than “heat can never pass from a cold to a hot body without some other change” that was only viewed by Clausius as a natural phenomenon. Here, we propose the theorem of the equivalence of transformations for reversible thermoelectric cycles. The analysis shows that the supercooling phenomenon Schilling et al. observed is achieved by a reversible combined power–refrigeration cycle. According to the theorem of equivalence of transformations in reversible thermoelectric cycles, the reduction in body temperature to below the ambient temperature requires the body itself to have a higher initial temperature than ambient as compensation. Not only does the supercooling phenomenon not bend the second law, but it provides strong evidence of the second law.

Thermodynamics contains rich symmetries. These symmetries are usually considered independent of the structure of matter or the thermodynamic state where matter is located and, thus, highly universal. As Callen stated, the connection between the symmetry of fundamental laws and the macroscopic properties of matter is not trivially evident. However, this view is now being challenged. Recently, with symmetry to the ideal gas equation of state (EOS), an ideal dense matter EOS has been proposed, which has been verified to be in good agreement with the thermodynamic properties of high-density substances. This indicates that there is a certain symmetry between the thermodynamic properties of substances in their high- and low-density limits. This paper focuses on the distinctive features and the significance of this symmetry. It is a new class of symmetry that is dependent on the thermodynamic state of matter and can be incorporated into the existing symmetrical theoretical system of thermodynamics. A potential path for developing the EOS theory arising from this symmetry is discussed. EOS at high densities could be developed by correcting or extrapolating the ideal dense matter EOS based on this symmetry, which might fundamentally solve the difficulty of constructing EOS at high densities.

Unlike the existing techniques of heat transfer enhancement using extended surfaces and/or turbulence promoters etc., the present study starts from a revisit at the mechanism of convective heat transfer, and then, indicates that the heat transfer rate depends not only on the flow and temperature fields, but also on their coordination. Based on the analyses and numerical and experimental verification of a number of convection cases, we suggest a principle called the principle of field-coordinated enhancement of convection ----- the better the coordination between the flow and heat flow fields, the higher is the heat transfer rate of convection. This principle can lead to novel approaches for heat transfer enhancement, which can markedly enhance heat transfer, but have less additional increase in the flow resistance. The field coordination number, Fc, which is equal to the integral of the dot product of the dimensionless velocity and temperature gradient vectors over the domain studied, has been introduced to describe the coordination degree of the velocity field and temperature gradient field for thermal convection. Hence, The field coordination number can act as criteria to compare the heat transfer performance of different convection cases.

Clausius entropy is a core concept of thermodynamics. The derivation of the Clausius entropy expression is based on the assumption of an ideal gas, so it is necessary' to verify the adaptability of the Clausius entropy expression in the real gas. Although the theoretical method can prove that the Clausius entropy expression is equally applicable to any real gas, there are still some shortcomings. For example, it needs to be proved by the circulation of the ideal gas with the real gas, rather than by the entropy change of the thermodynamic process, and the microphysical mechanisms of the thermodynamic parameter relationships of the real gas that differ from those of the ideal gas cannot be analyzed. However. most extant literature or textbooks do not explain this and instead apply the expression of entropy based on the assumption of an ideal gas to the real gas, which is insufficiently rigorous. Therefore. it is necessary to apply molecular dynamics simulation to provide a direct proof of Clausius entropy expression by simulating the thermodynamic processes of real gases. Firstly, the temperature, pressure, and density obtained from simulations during the isothermal process are compared with data from the National Institute of Standards and Technology (NIST) to verify the accuracy of the molecular dynamics simulation under high pressures. The isothermal processes of argon and carbon dioxide at high pressures are simulated. and the relative errors are within 8% compared with the NIST data, thus demonstrating the accuracy of the calculations. In addition_ the reversibility of the simulated process is verified in this work by comparing the thermodynamic results of the adiabatic compression process with those of the isentropic process, and when the input work at each step of the adiabatic compression process is small enough, the calculated results are very close to the isentropic line. which indicates that the thermodynamic process of simulation is approximately reversible. Thirdly. the heat temperature quotients of isochoric process are calculated, and the results are compared with the corresponding entropy changes in the NIST (the relative errors are less than 5%), thus directly verifying the applicability of the Clausius entropy expression to real gases. Then, the thermodynamic cycles of argon and carbon dioxide under high pressures are simulated to further check whether the Clausius entropy expression is relevant to the real gas by modeling whether the sum of heat and temperature quotients of the thermal cycle process is zero. The results show that the sum of the heat temperature quotients of the cycles is extremely small. which further verifies that the Clausius entropy expression is also applicable to the real gas. Finally, the microscopic analysis indicates that the relationship of thermodynamic parameters in real gas thermodynamic processes differs from the physical mechanism of an ideal gas because of the interconversion of molecular kinetic energy and molecular potential energy. Obviously, when the system pressure is small, the conversion between molecular kinetic energy and molecular potential energy inside the ideal gas system is small, and the variation of heat capacity is negligible; when the system pressure increases, the conversion between molecular kinetic energy and molecular potential energy inside the system increases. and the variation of heat capacity increases accordingly. Since this interconversion occurs within the system, the Clausius entropy expression remains unchanged.

The motion of heat can be described by a thermal energy–momentum tensor. A general description of heat conduction in the geometrical language is developed based on a two-phase continuum model within the framework of relativistic continuum dynamics, and the momentum balance equation of heat, or the general heat conduction equation (GHCE), is derived by the balance equation of the thermal energy–momentum tensor. We demonstrate that the low-speed limit of the GHCE coincides the former version of GHCE of the thermomass theory that is based on the energy–mass equivalence and Newtonian dynamics. Numerical solutions of the GHCE are also provided to demonstrate that the GHCE not only overcomes the paradox of instantaneous heat propagation of the Fourier's law, but also resolves the defect of the CV model that temperatures may drop below absolute zero under some conditions. The present work formulates a general description of the thermal transport processes, and can provide a deeper insight into the heat transfer discipline from the relativistic point of view.